The influence of experimental and sampling uncertainties on the probability of unsatisfactory performance in geotechnical applications

Transcription

1 The influence of experimental and sampling uncertainties on the probability of unsatisfactory performance in geotechnical applications D. Mašín Charles University in Prague Faculty of Science Albertov Prague 2, Czech Republic masin@natur.cuni.cz Tel: , Fax: Number of words, main text: 479 Number of words, total: 587 Number of tables: Number of figures: 6 August 23, 25 Manuscript accepted for publication in Géotechnique

2 Abstract While the effect of spatial variability on the probability of unsatisfactory performance in geotechnical applications is relatively well understood, comparatively less attention has been given in the literature to the effects of experimental (measurement scatter) and sampling (insufficient number of samples) uncertainties. In this paper, a general approach is developed to incorporate experimental and sampling uncertainties into probabilistic analyses based on random field methods. It is shown that, when compared with the standard approach which attributes the measured total soil variability to spatial variability, consideration of experimental uncertainty may significantly reduce the calculated probability of unsatisfactory performance. It is argued that this may be one of the reasons for an overestimation of the probability of unsatisfactory performance in geotechnical probabilistic simulations (another important reason is the spatial averaging of soil properties). Evaluation of the sampling uncertainty reveals that, although a relatively large number of samples is needed for spatial variability characterisation, a limited number of samples is sufficient to quantify the experimental uncertainty. It is pointed out that no adjustments of the existing random field-based software are needed to consider the additional uncertainties. To illustrate the proposed approach, two extensive experimental data sets on sand are presented: one reflecting total variability and the other quantifying experimental uncertainty. A hypoplastic model is calibrated against the two data sets and adopted in random field analyses of strip footing settlement. Keywords: probabilistic methods; random field method; experimental uncertainty; sampling uncertainty; constitutive model; sand. Introduction Design parameters obtained from any geotechnical site investigation are subject to uncertainties. These are caused, in particular, by (a) an inherent spatial variability of soil properties, (b) experimental uncertainty (measurement scatter) due to limitations of the experimental techniques and (c) sampling uncertainty (statistical uncertainty) due to the limited number of soil samples used in the investigation (Phoon & Kulhawy (999a), Schweiger & Peschl (25), Christian et al. (994)). Other uncertainty sources, which are not within the scope of this paper, involve transformation uncertainty (transformation of the measured value to the design value by means of a potentially inaccurate empirical method), material model uncertainty (imperfect representation of soil behaviour by a material model) and experimental bias (systematic measurement error due to problematic functioning or calibration of the experimental device). Formal treatment of the spatial variability is relatively well understood, as it has been studied

3 thoroughly in the geotechnical literature. Spatial variability has been investigated experimentally (e.g., Suchomel & Mašín (2), Zhao et al. (23), Akbas & Kulhawy (2), Wang & Chiasson (26), Hicks & Onisiphorou (25), Kim (22), Lumb (966), Soulié et al. (99), Jung et al. (28), El-Ramly et al. (25), El Gonnouni et al. (25), Elkateb et al. (23)), treated formally using random-field methods (e.g., Vanmarcke (983), Griffiths & Fenton (24), Suchomel & Mašín (2), Suchomel & Mašín (2), Nübel & Karcher (998), Andrade et al. (28), Cho & Park (2), Niemunis et al. (25), Kim & Santamarina (28), Hsu & Nelson (26)) or by using approximate probabilistic methods involving spatial averaging of soil properties (e.g., Suchomel & Mašín (2), Suchomel & Mašín (2), Brza kała & Puła (996), Peschl & Schweiger (23), Cho (27), El-Ramly et al. (22)). It has been shown that the spatial correlation significantly affects the results of probabilistic analyses of various geotechnical problems (e.g., Wang & Chiasson (26), Hicks & Onisiphorou (25), Nour et al. (22), Fenton & Griffiths (22), Fenton & Griffiths (25), Huber et al. (2), Griffiths et al. (22), Fenton & Griffiths (23), Hicks & Samy (22), El-Ramly et al. (26), Haldar & Babu (28), Popescu et al. (997)). Less attention has been payed to the other uncertainty sources. An additive model separating different uncertainties has been proposed by Phoon & Kulhawy (999b). They considered the reduction of spatial variability using a spatial averaging factor (Vanmarcke, 983) and additively included variance due to experimental uncertainty in the global parameter variance. However, this may lead to an overestimation of the global uncertainty; this issue is discussed in detail later in this paper and was also implied by Müller et al. (24). A similar simplified approach has been followed by Akbas & Kulhawy (2) and by Al-Naqshabandy & Larsson (23), who also included the effects of sampling and transformation uncertainties. A general approach to uncertainty quantification, including a proper treatment of the experimental uncertainty, was presented by Christian et al. (994) and Müller et al. (24); they did not consider random field characterisation of spatial variability, but instead adopted a simpler method based on spatial averaging. Another approach to quantify the effect of experimental uncertainty is represented by the theory of random sets, applied to geomechanical problems by Schweiger & Peschl (25) and Oberguggenberger & Fellin (28). More publications on separating different uncertainty sources can be found in the field of hydrology (Merz & Thieken (25), Sun et al. (22), Booy & Lye (989)) and other engineering disciplines (Castrup (24), Frey (993), Helton (997)). From an experimental perspective, experimental uncertainty can, in principle, be estimated by quantifying the variations in repeated measurements of the same property on identical soil samples. As pointed out by Akbas & Kulhawy (2), however, very few studies of this kind can be found in the geotechnical literature. Statistical evaluation of experiments on a relatively uniform m clay sample was presented by Watabe et al. (27). An experimental characterisation of sampling uncertainty has been presented by Bourdeau & Amundaray (25) and Wu et al. (23). In this paper, a general approach is developed to incorporate spatial variability and experimental 2

4 and sampling uncertainties into geotechnical analyses based on the random field method. Subsequently, extensive experimental data sets on both the spatial variability and experimental uncertainty are presented and the proposed approach is demonstrated by simulation of a typical geotechnical problem; the settlement of a strip footing. Incorporation of experimental uncertainty In this section, it is assumed that the global variability and the experimental uncertainty are known exactly. That is, the sampling uncertainty is neglected, which will be covered later. It is assumed that the following two measurement data sets are available: Measurements in the spatial grid aimed at evaluating spatial variability for random field probabilistic analyses. These measurements contain information on the spatial variability. Inevitably, however, the data from each spatial node are themselves subject to the experimental uncertainty. Repeated measurements on nominally identical samples to quantify the experimental uncertainty. Let us denote the uncertain parameters asx i (whereiis the parameter number). X,X 2...X n may represent, for example, parameters of the constitutive model (such as friction angle and cohesion), or uncertain values of the state variables (such as the void ratio). Calculation of spatial variability by extracting the experimental uncertainty from the total variability The value of the parameter X i, which depends on spatial position x, can be written as X i (x) = µ[x i ](x)+ε t [X i ](x) () where µ[x i ](x) is the mean value of the parameter X i at the given position x and ε t [X i ] is the stochastic component which lumps different sources of uncertainty (subscript t stands for total ). In the following, the spatial variation is considered a spatially random process without any trend in the dependency of µ[x i ] by location. That is, the mean value µ[x i ] is considered as a constant, and the total variability component is represented by ε t [X i ] with zero mean value. Thus, X i (x) = µ[x i ]+ε t [X i ](x) (2) 3

5 The total variability is (in this section) composed of two components: spatial variability ε n (subscript n for natural ) and experimental uncertainty ε e (subscript e for experimental ). ε n depends on the spatial location x, whereas the experimental uncertainty is assumed to be independend of location. Therefore: ε t [X i ](x) = ε n [X i ](x)+ε e [X i ] (3) and X i (x) = µ[x i ]+ε n [X i ](x)+ε e [X i ] (4) As the mean value of X i (x) is constant, and ε n [X i ](x) and ε e [X i ] are independent variables, variance of X i (x) is calculated as: var[x i (x)] = var[ε t [X i ](x)] = var[ε n [X i ](x)]+var[ε e [X i ]] x R (5) where R is the area of interest, representing, for example, a geological layer with given parameter statistics. For brevity, statistical variables in this section always represent parameters X i and the notation can be simplified asε t [X i ] = ε t. Therefore: var[ε t (x)] = var[ε n (x)]+var[ε e ] x R (6) which allows us to calculate the variance due to the spatial variability from the known total variance and variance due to the experimental variability as var[ε n (x)] = var[ε t (x)] var[ε e ] x R (7) For text brevity, the spatial position identified x will be omitted in the following text. Quantification of spatial correlation properties for zero experimental uncertainty Probabilistic analyses that consider the spatial variability of material properties often make use of so-called random field methods. These methods are based on spatial statistics of the input parameters, which are adopted to generate a random field. Statistical characteristics of the performance function (model output) can then be evaluated using methods such as Monte-Carlo, which involve a number of random field realisations. A random field can be characterised by the statistical distribution of the input parameter and by the dependency between measurements of the parameter at two locations on the separation distance. This dependency may be described by a semi-variogram γ v (τ). An empirical semi-variogram can To avoid confusion with the variance reduction factor γ, the semi-variogram is in this paper denoted as γ v in place of the standard notationγ. 4

6 v ( ) var[ t ] or ( ) v ( )/var[ t ] ( ) Figure : Normalised semi-variogram and correlogram for zero experimental uncertainty (var[ε e ] = ). be constructed from measurements of the i-th parameter X i using γ v (τ) = 2 E [(X i (x ) X i (x 2 )) 2], x x 2 = τ (8) where τ is the spatial distance between two points and E is the expected value operator (mean). For spatialy limited correlation, lim τ γ v (τ) represents the variance of the random field: lim γ v(τ) = var[ε t ] (9) τ In the geomechanical literature, it is more common to express the semi-variogram in the normalised form using the autocorrelation coefficient ρ(τ) (in this graph it is denoted as a correlogram), which, for zero experimental uncertainty var[ε e ] =, reads ρ(τ) = γ v(τ) var[ε t ] () The correlogram may be modelled in different ways; in geomechanical literature the Markov exponential model is popular (Vanmarcke (983), Griffiths & Fenton (24)): ( ) 2τ ρ(τ) = exp θ with the parameter θ denoted as the correlation length. Both the semi-variogram (normalised by var[ε t ]) and correlogram for zero experimental uncertainty var[ε e ] = are shown in Fig.. () Quantification of spatial correlation properties for non-zero experimental uncertainty Spatial correlation properties for non-zero experimental uncertainty can be expressed using a total semi-variogram (Smith, 24), which includes both the natural spatial variability and the experimental uncertainty. Such a semi-variogram is depicted in Fig. 2, together with three characteristics denoted in statistics as nugget, range and sill. They are equivalent to var[ε e ], correlation 5

7 γ v (τ) sill = var[ε t ] γ v (τ) nugget = var[ε e ] τ range = θ Figure 2: Total semi-variogram for non-zero experimental uncertainty. v ( ) var[ε n ] or ρ(τ) γ v (τ) ε n ] ρ(τ) γ v (τ) var[ε e ] or ρ(τ) γ v (τ)/var[ε e ] ρ(τ) (a) Figure 3: (a) Normalised semi-variogram and correlogram for natural spatial variability, (b) discontinuous normalised semi-variogram and correlogram for experimental uncertainty. (b) τ length θ and var[ε t ], respectively, using the terminology adopted in this paper. For the purpose of introducing the spatial statistical analysis into a code with non-zero element size, however, the developments are not based on the total semi-variogram from Fig. 2. Instead, separate variograms (or correlograms) are considered for natural variability and for experimental uncertainty. The semi-variogram for natural variability is formally the same as the semi-variogram introduced in the previous subsection (Fig. ), normalised with respect tovar[ε n ] instead ofvar[ε t ]. var[ε n ] is calculated using Eq. (7). This semi-variogram is shown in Fig. 3a. The semi-variogram for the experimental uncertainty is different; since the experimental uncertainty is assumed to be spatially independent (Eq. (3)), its semi-variogram and correlogram are discontinuous (Fig. 3b), with perfect correlation for τ = and zero correlation for τ >. Spatial correlation for analysis with elements of finite size When random field analysis is incorporated into the finite element method (or any other method with non-zero size of calculation units, denoted as elements in finite element and difference methods), the following problem must be considered. Spatial characteristics described in the previous sections are relevant for calculation units of infinitesimal size. For elements of finite size, the 6

8 variability of parameters (measured by their variance) must be reduced, due to the averaging of property values within the finite element. This spatial averaging of soil properties over the finite element is denoted as local averaging (Vanmarcke, 983). The variance reduction factor γ is for zero experimental uncertainty calculated as: var[ε t ] A = γvar[ε t ] (2) wherevar[ε t ] A is the variance of the spatially-averaged random field which enters the random field simulations instead of var[ε t ]. The variance reduction factor γ is for the correlogram described using Eq. () and for a 2D square finite element calculated by integration of () as (Vanmarcke, 983): γ = 4 αθ αθ ( (αθ) 4 exp 2 ) x θ 2 +y 2 (αθ x)(αθ y)dxdy (3) where x, y are local spatial coordinates within the element frame of reference, θ is the correlation length and α = a/θ is the element size factor, withabeing the size of the square finite element. For the random field simulations with non-zero experimental uncertainty, the separate semi-variograms shown in Fig. 3 are used. The variance reduction factorγ n for spatial variability is calculated using Eq. (3). Spatial variability is then introduced by reducing the variance var[ε n ] calculated using (7) instead of reducing var[ε t ] as is done in standard random field simulations, which neglect the experimental uncertainty. Thus, var[ε n ] A = γ n var[ε n ] = γ n (var[ε t ] var[ε e ]) (4) The variance reduction factor for the experimental variability γ e comes from the integration of the discontinuous correlogram from Fig. (3b). It follows that for the non-zero element size γ e is equal to. That is, experimental variability enters the random field simulations only through the quantification of var[ε n ] from the known var[ε t ] and var[ε e ]. Since the variance reduction factor γ n is the same in the analyses neglecting and considering the experimental uncertainty, the incorporation of the experimental uncertainty into random field simulations requires no adjustments of the existing software. The total parameter variance var[ε t ] must only be replaced byvar[ε n ]. Implications of neglecting the experimental variability The above analysis implies a possibly counter-intuitive effect of experimental variability. The increase of experimental uncertainty for the given total variability measured in the spatial grid using non-perfect experimental methods decreases the parameter variance that should enter random field simulations. Consequently, it also reduces the variance of the performance function. Statistics of 7

9 Rel. frequency PDF considering experimental uncertainty PDF neglecting experimental uncertainty design value (minimum) performance function Figure 4: Sketch of the influence of experimental uncertainty on the calculated probability of unsatisfactory performance p f (sketched for the case when the performance is unsatisfatory for the performance function lower than the design value, applicable for example for factor of safety). the performance function may be used to quantify the probability of unsatisfactory performance p f (Duncan, 2). It represents the area under the probability density function (PDF) for calculated values that are lower than the limiting design value (in the case when the design value is the minimum, for example factor of safety), or it represents the area under the PDF for calculated values that are higher than the limiting design value (in the case when the design value is the maximum, for example foundation settlement or lining bending moment). The effect of experimental variability is demonstrated in Fig. 4. It follows that for p f < 5% consideration of experimental uncertainty decreases the probability of unsatisfactory performance. A specific example is given later in this paper. Experience shows that probabilistic methods in geotechnical engineering overpredict the probability of unsatisfactory performance. For example, Griffiths & Fenton (24) noted that, for the case they studied, a direct conversion of the factor of safety FS =.47 into the probability of slope failure without considering spatial variability yielded p f = 28 %, a remarkably high value considering that slopes with FS =.47 rarely fail. It is now well established that this discrepancy may be explained by means of spatial averaging (Griffiths & Fenton, 24). In this paper, it is argued that overprediction of the probability of unsatisfactory performance may additionally be caused by neglecting the experimental variability in random field simulations. Incorporation of sampling uncertainty At this point, let us continue our analysis by considering the sampling uncertainty, which is the uncertainty derived from the fact that the total and experimental uncertainties have been evaluated 8

10 using an insufficient number of samples. Quantification of sampling uncertainty All the directly measured statistical characteristics involved in the simulations are subject to sampling uncertainty, that is µ[x i ], var[ε t [X i ](x)] and var[ε e [X i ]] (abbreviated as µ, var[ε t ] and var[ε e ]). In this section, these quantities become random variables. For clarity, their values obtained by evaluation (of a finite number) of experiments are distinguished using a circumflex accent σ. In the following, n t denotes the number of samples used in the evaluation of the spatial variability and n e is the number of samples used in the evaluation of the experimental uncertainty. For normally distributed parameters (Hald (95), Booy & Lye (989)) σ s [ µ] = σ[ε t] nt (5) σ 2 s σ 2 s [ σ 2 [ε t ] ] = 2 σ4 [ε t ] n t [ σ 2 [ε e ] ] = 2 σ4 [ε e ] n e (6) (7) where the subscript s is adopted to indicate sampling uncertainty. Note that Eqs. (5) to (7) are approximate only, as the sampling uncertainty should be calculated from the exact values of σ[ε t ] and σ[ε e ] (not from their experimental estimates σ[ε t ] and σ[ε e ]). Also, they are valid for noncorrelated variables only (this is satisfied for experimental uncertainty, but not for total uncertainty with distance-dependent autocorrelation coefficient ρ). To incorporate the sampling uncertainty into the random field simulations with a calculation unit of finite size, the sampling standard deviation of σ[ε n ] (denoted as σ s [ σ[ε n ]]) must be quantified. It follows from the rules of statistics that for two uncorrelated normally distributed variablesx and Y var[x Y] = var[x]+var[y] (8) Therefore, from (6) and (7): σ 2 s [ σ 2 [ε n ] ] = 2 σ4 [ε t ] n t + 2 σ4 [ε e ] n e (9) At this point, the following rule of statistics is employed: σ 2 [X 2 ] = 2σ 2 [X] ( σ 2 [X]+2µ 2 [X] ) (2) 9

11 to quantify σ s [ σ[ε n ]]. A quadratic equation is obtained with two roots: one negative and one positive. As standard deviation must be positive, the equations yield a single valid solution, which after algebraic manipulations reads: σ s [ˆσ[ε n ]] = where σs 2 [ σ 2 [ε n ] ] is given by (9). σ 2 [ε n ]+ σ 4 [ε n ]+ σ2 s [ σ 2 [ε n ]] 2 (2) Incorporation of sampling uncertainty into the random field method The most general approach for quantifying sampling uncertainty within a random field method is the two-dimensional Monte-Carlo simulation (Hofer et al., 22). In this method a nested Monte- Carlo simulation is run for each of the random field realisations, keeping the same random field structure (the random field is defined by means of normalised random variables, which do not change within the nested Monte-Carlo run). µ[x i ] and σ[ε n [X i ]] in the nested Monte-Carlo loop are considered as random variables with mean values µ[x i ] and σ[ε n [X i ]] and standard deviations σ s [ µ[x i ]] (Eq. (5)) and σ s [ σ[ε n [X i ]]] (Eq. (2)), respectively. Formally, 2n random variables are thus involved in the nested Monte-Carlo loop (n is the number of parameters). The two-dimensional Monte-Carlo simulation is computationally extremely demanding. To simplify it, let us formally describe the computational model as Y = g(x,x 2,...,X n ) (22) wherey is the performance function (simulation result, such as the factor of safety). In the simplified analysis, the mean value and variance of Y due to natural variability considering the effect of experimental uncertainty (denoted as µ n [Y] and var n [Y]) may be found using random field analysis with parameter statistics µ[x i ] and σ[ε n [X i ]]. Next, a single random field realisation is selected (preferably the one leading toy µ n [Y]) and a single Monte-Carlo simulation is run with a fixed random field structure that has variable µ[x i ] and σ[ε n [X i ]] using their mean values µ[x i ] and σ[ε n [X i ]] and standard deviations σ s [ σ[ε n [X i ]]] (Eq. (2)) andσ s [ µ[x i ]] (Eq. (5)) leading to the performance function statistics µ s [Y] and var s [Y]. The assumption of statistical independence of sampling uncertainty and natural variability then yields var[y] = var n [Y]+var s [Y] (23) µ[y] = µ n [Y] (24)

12 To further simplify the simulations, it is pointed out that the Monte-Carlo run for sampling uncertainty does not involve variation of the random field structure. It may thus be quite reliably substituted by simpler probablisitic methods such as the first order second moment (FOSM) method or one of the point estimate methods (Christian & Baecher (999), Rosenblueth (98), Zhou & Nowak (988)). When expressed using the FOSM method, the value of var s [Y] reads var s [Y] = 4 n FOSM j= [g(µ[r j ]+σ[r j ]) g(µ[r j ] σ[r j ])] 2 (25) wherer j formally denotes the uncertain variables (µ[x i ] andσ[ε n [X i ]]). It follows thatn FOSM = 2n (standard deviation and mean of each parameter X i is involved as an uncertaint variable). The FOSM method therefore requires 4n runs of the performance function, a comparatively lower number than the potentially thousands of runs required by the Monte-Carlo method. Geotechnical engineering example In this section, an example of application of the proposed approach is given. The underlying case has been thoroughly described in Suchomel & Mašín (2). They studied the total variability of a sandy soil using experiments on samples obtained from a regular grid in a quarry wall. The results have been used in simulations of a strip footing settlement using a hypoplastic model. In this paper, an additional experimental data set is presented quantifying the experimental uncertainty and its effect is evaluated using random field simulations. Experimental data quantifying natural variability The soil studied comes from the upper Cretaceous basin in south Bohemia (Czech Republic). The sediments of the basin are fluvial, characterised by a variation of gravely sands, sands and clayey sands. As these materials can all be modelled using a single constitutive model (hypoplastic model has been selected), where the variation in granulometry is represented by the parameter variability, the case study is suitable for utilisation of probabilistic methods based on random fields. For material model calibration, thirty-seven samples for testing total variability were extracted from the quarry wall in a grid depicted in Fig. 5 (three out of forty samples shown in Fig. 5 could not be used in the analyses due to experimental problems). It is to be pointed out that the samples extracted from the quarry wall have been disturbed; we have thus focused on quantification of variability due to soil composition reflected by the hypoplastic model parameters, not on possible variability of in-situ void ratio. The testing programme consisted of the following experiments:

13 . Angle of repose test 2. Oedometric test. The oedometric specimens were prepared by air pluviation of the dry sand without any compaction and subsequent flooding. The aim was to prepare specimens of minimum relative density which are most suitable for asymptotic state quantification (Mašín, 22). 3. Triaxial test. The triaxial specimens were prepared by a spooning of saturated samples directly into the triaxial mold of the diameter of 38 mm, using the procedure described in Tatsuoka et al. (979) and Wichtmann & Triantafyllidis (22). As noted by Head (985), saturated spooning is preferable to placing the sample dry and subsequent flooding, as it is difficult to ensure full saturation in the latter case. The sample has been prepared to the total height of 78 mm in five layers, with each layer having been thoroughly densified by tamping (Tatsuoka et al., 979) using rod of 2 mm in diameter. As the parameters of the hypoplastic model depend on granulometry only and not on relative density (Herle & Gudehus (999), Hájek et al. (29)), the tamping energy has not been controlled precisely; instead, approximately the same tamping energy has been applied on all the samples by the experimentalist. After the sample preparation, it has been consolidated isotropically up to the effective stress of 2 kpa (under a backpressure of 2 kpa) and sheared drained up to failure. Figure 6a shows summary of the all angle of repose test results, Figures 7a and 8a give a summary of the triaxial test results and Figure 9a shows all the oedometric curves. Summary graphs are presented only for brevity, without any indication of location of the individual samples within the sampling grid; for detailed information, the readers are referred to Suchomel & Mašín (2). Note that the sample preparation method probably contributed to the variations in the initial void ratio; however, it should not have a major influence on the calibration results, as the initial void ratio does not affect substantially the hypoplastic model parameters. Suchomel & Mašín (2) used the experimental results to calibrate the hypoplastic model given by von Wolffersdorff (996). The model requires eight parameters: ϕ c, h s, n, e c, e d, e i, α and β. A detailed description of the model parameters and their calibration procedures can be found in von Wolffersdorff (996) and Herle & Gudehus (999); here we present their summary only for brevity: ϕ c is the critical state friction angle. Parameters h s, n, e c, e d and e i define the limiting void ratio lines, prescribed by Bauer (996) [ ( ) 3p n ] e = e p exp h s (26) 2

14 Figure 5: The wall of the sand pit indicating positions of specimens for investigation of total variability (from Suchomel and Mašín 2). Note that three samples e4, f and f2 could not be tested successfully. Rel. frequency φ c [ ] Rel. frequency φ c [ ] (a) Figure 6: Normalised histograms and Gaussian fits for the angle of repose test. (a) experiments on total variability, (b) experiments on experimental uncertainty. (b) where p is the mean effective stress and e p is substituted by e d for the minimum void ratio, e c for the critical state void ratio and e i for the theoretical state of minimum relative density. Parameter α controls the dependency of peak friction angle on relative density (increase of α increases the peak friction angle for the given relative density). The model automatically predicts an increase of peak friction angle with increasing relative density. Parameter β controls the dependency of soil stiffness on relative density and mean effective stress (increase of β increases stiffness for the given relative density and mean effective stress). The model automatically predicts an increase of soil stiffness with increasing relative density and with increasing stress. 3

15 8 8 q [kpa] q [kpa] natural variability - experiments 2 experimental uncertainty - experiments εs [-] natural variability - simulations 4.8 εs [-]..2.4 (b) q [kpa] q [kpa] (a).6 6 experimental uncertainty - simulations εs [-]..2.4 (c) εs [-]..2.4 (d) Figure 7: Stress-strain curves of drained triaxial tests. Experimental curves of the tests on total variability (a) and experimental uncertainty (b); simulated curves of the tests on total variability (c) and experimental uncertainty (d). The parameters were calibrated as follows. ϕc was measured directly using the angle of repose test (Figure 7a). hs, n and ec were calibrated using the oedometric tests, such that they can be represented using equation (26) where ep ec was assumed. The mean effective stress p has been calculated from the vertical stress σv using the formula by Ja ky (944) leading to p = σv ( 2 sin ϕc /3). ed and ei were estimated from ec using the empirical correlations ed =.38ec, ei =.2ec (Suchomel & Mas ı n, 2). The parameters α and β were calibrated using the triaxial test results. α was calibrated so that the model represented the exact peak friction angle and β was calibrated to predict the shear stiffness at q =.5qmax, where qmax is the peak deviator stress. All the parameters were calibrated using automated scripts to limit ambiguity of calibration. Note that the oedometric experiments conducted by Suchomel & Mas ı n (2) were performed up to vertical stresses of 64 kpa, whereas the new data set on experimental uncertainty was performed up to 32 kpa. For consistency, the model was re-calibrated (loading step 32 kpa to 64 kpa was not considered) and the parameters thus differ slightly from those presented in Suchomel & Mas ı n (2). 4

16 .5.45 natural variability - experiments.5.45 experimental uncertainty - experiments.4.4 e [-].35 e [-] ε s [-] ε s [-] (a) (b).5.45 natural variability - simulations.5.45 experimental uncertainty - simulations.4.4 e [-].35 e [-] ε s [-] ε s [-] (c) Figure 8: Void ratio variation curves from drained triaxial tests. Experimental curves of the tests on total variability (a) and experimental uncertainty (b); simulated curves of the tests on total variability (c) and experimental uncertainty (d). (d) Figures 7c, 8c and 9c show the simulations of the laboratory experiments with the obtained parameter sets, indicating a reasonably good fit. The oedometric curves (Figure 9) have been obtained by directly plotting Eq. (26), whereas the triaxial curves (Figure 7) have been obtained by single element simulations of the experiments using the hypoplastic model. Histograms of the parameters with Gaussian fits are shown in Fig. a (logarithms of parameter values approximated in the case of log-normally distributed parameters); the parameters of the distributions are in Table. Some parameters were more closely approximated by a log-normal distribution and some by the Gaussian distribution. The more suitable distribution was selected using the Kolmogorov-Smirnov test. 5

17 e [-].7.6 e [-] natural variability - experiments.3 p [kpa] (a).4 experimental uncertainty - experiments.3 p [kpa] (b) e [-].7.6 e [-] natural variability - simulations.3 p [kpa] (c).4 experimental uncertainty - simulations.3 p [kpa] Figure 9: Oedometric test results. Experimental curves of the tests on total variability (a) and experimental uncertainty (b); simulated curves of the tests on total variability (c) and experimental uncertainty (d). (d) 6

18 Rel. frequency ln(h s / kpa) Rel. frequency ln(h s / kpa).4 3 Rel. frequency Rel. frequency ln(n) ln(n) Rel. frequency e c Rel. frequency e c Rel. frequency ln(α) Rel. frequency ln(α) Rel. frequency (a) β Rel. frequency Figure : Normalised histograms and Gaussian fits of hypoplastic model parameters. (a) total variability, (b) experimental uncertainty. 7 (b) β

19 Table : Characteristic values of statistical distributions of the hypoplastic model parameters (values in italics were obtained from empirical correlations rather than direct calibration). total experimental ratio parameter mean st. dev. mean st. dev. var[ε e ]/var[ε t ] ϕ c [ ] ln(h s /kpa) ln n e c e i e d ln α β Suchomel & Mašín (2) also evaluated the correlation length using the ϕ c measurements. The correlogram was represented by the Markov function () enhanced by considering different correlation lengths in the horizontal (θ h ) and vertical (θ v ) directions, which led to θ h = 242 m and θ v = 5. m. However, the estimation of horizontal correlation length was very approximate due to insufficient data in the lateral plane (Uzielli et al. (25), DeGroot & Baecher (993)), and the vertical correlation length estimation was also not reliable, as the vertical sampling distance of 3 m was too high to reveal the vertical scale of fluctuaction. For a more accurate estimation of vertical correlation length, additional angle of repose tests were performed using samples obtained from a 5m high profile with.5m vertical sampling distance, which led to a more precise estimation of θ v =.3 m. Suchomel & Mašín (2) subsequently conducted a numerical parametric study on the influence of correlation length on the simulation results. In this work, the investigation of the correlation length effect is not a scope of the paper; instead, we aim to demonstrate the effect of experimental and sampling uncertainties. As considering the low correlation length of θ v =.3 m requires very fine finite element meshes and computationally demanding simulations, we performed the simulations using fixed values of θ h = 242 m and θ v = 5. m, noting that for the present case study the adopted value of θ v is unrealistically high. The selection of θ v would affect the standard deviations of the performance function and thus also the calculated probability of unsatisfactory performance, but not the qualitative effect of the experimental and sampling uncertainties on the simulation results. 8

20 Experimental data quantifying experimental uncertainty A new set of experiments was performed to quantify the experimental uncertainty (Mayer, 23). This experimental programme was identical to that used to evaluate total variability; however, soil was obtained from a single location within the quarry wall 2. The experiments were performed on forty nominally identical samples, which were prepared from the main sample using the cone and quartering method. The main sample of approx. kg has first been mixed thoroughly. The technique then involved pouring the sample into a cone, flattening the cone, dividing the flattened cone into four equal divisions (quartering), removing two opposite quarters and repiling the other two into a new cone (Schumacher et al., 99). The experiments were performed in the same soil mechanics laboratory using the same triaxial and oedometric apparatus as the experiments aimed at quantification of the spatial variability. The two experimental data sets have been performed by two different experimentalists; however, one experimentalist performed all the tests within one experimental data set. The preparation procedure for the individual samples was identical to the preparation of the samples for testing spatial variability. Figure 6b shows results of the angle of repose tests, Figures 7b and 8b summarise the triaxial test results and Figure 9b shows all the oedometric curves. The results show a substantial scatter, in particular in the peak strength measured in the triaxial test and in the initial void ratio of both the oedometric and triaxial tests. Visual comparison with the tests on total variability (Figs. 7a, 8a and 9a), however, reveals that these experimental results are even more variable. Figures 7d, 8d and 9d show simulations of the experiments using the hypoplastic model. A reasonably good approximation is obtained, indicating that the statistics of the model parameters may be considered as representative for the uncertainty evaluation. Histograms of the hypoplastic model parameters with Gaussian fits are shown in Fig. b (log-transformed parameters were fit in the case of lognormally distributed parameters), and the parameters of the distributions are given in Tab.. Tab. also gives the ratios of parameter variances of data sets on total variability and experimental uncertainty. Lower values of this ratio (indicating relatively low experimental variability) apply to the angle of repose (ϕ c ) and parameters controlling the slope of the oedometric curve (h s and n). In contrast, higher values were obtained for the initial void ratio measurements (e c ) and for parameters controlling the triaxial stress-strain curves α and β. In fact, var[ε e ]/var[ε t ] for the parameter α is higher than one, which is theoretically not possible and other effects not considered in this paper probably play a role here (for example uncertainty in the model calibration, different sampling location for the tests on experimental uncertainty, etc). 2 Note that due to the quarry wall advance between the sample collection, the new sample could not be extracted exactly from the section shown in Fig. 5. It was, however, ensured the new sample had a geologically equivalent position to the samples obtained earlier. 9

21 Description of the finite element model The influence of the experimental and sampling uncertainties was investigated by the random field simulation of the settlement of a strip footing. The simulated problem aimed to demonstrate the concept and did not correspond to any real case study. The problem characteristics were taken from Suchomel & Mašín (2). Simulations were performed using a finite element package Tochnog Professional (Rodemann, 28). The problem geometry and the finite element mesh are shown in Figure. The mesh consisted of 92 nine-noded quadrilateral elements. The foundation was analysed as rigid and perfectly smooth. The soil unit weight was 8.7 kn/m 3. The initial earth pressure coefficient at rest was K =.43 (calculated from the Jáky (944) formula using an average value of ϕ c ). The initial void ratio e =.48 was used in the simulations. The random fields of the input variables were generated using a Cholesky decomposition of the correlation matrix (e.g., Fenton (997)). All the parameters have been considered as uncorrelated, with the exception of e c, e d and e i, which were perfectly correlated (this is because e d and e i were calculated from e c using an empirical relationship). Cross-correlation between the individual parameters has been investigated, with statistically significant cross-correlation only being observed between the parameters α and β (R 2 =.6), α and e c (R 2 =.7) and α and ϕ c (R 2 =.5) (the cross-correlation matrix is presented in Suchomel & Mašín (2)). However, the effect of parameter correlation has not been studied within this paper and the correlations have thus been neglected. In the simulations, foundation displacements u y corresponding to the load of kpa were evaluated. The probability of unsatisfactory performance was calculated with respect to foundation settlements of.9 m. 5 m foundation 25 m 55 m Figure : Finite element mesh. 2

22 The influence of experimental uncertainty on the probability of unsatisfactory performance The influence of experimental uncertainty was included using Eq. (4). The Monte-Carlo method was used to quantify the statistics of the performance function. To ensure the accuracy of the simulations the number of realisations was always larger than 5; this is significantly higher than the minimum number of realisations (7) identified by Suchomel & Mašín (2)). The simulations were run for the total variability and experimental uncertainty obtained from the experiments. In addition, the influence of experimental uncertainty was further studied by varying it using a multiplier m e such that var[ε n ] A = γ n var[ε n ] = γ n var[ε t ] m e var[ε e ] (27) The value of m e = thus signifies a standard random finite element simulation, which does not consider the effect of experimental uncertainty. The value of m e = implies parameter statistics obtained from the present experiments. < m e < means experimental uncertainty lower than observed in the present experiments and m e > means experimental uncertainty higher than observed in the present experiments. var[ε n ] equal to zero was assumed for negative values of (var[ε t ] m e var[ε e ]) (the Macaulay bracket x in Eq. (27) gives a positive part of x, i.e. x = (x+ x )/2). Fig. 2 shows the random fields of the parameters h s, n and β for one of the Monte-Carlo realisations withm e =, which yielded u y very close to the median value ofu y obtained in the complete Monte-Carlo run. Suchomel & Mašín (2) have shown, using sensitivity analysis, thath s,nand β were the most influential parameters in these simulations. The value of the foundation vertical displacement closely matched a log-normal distribution. An example simulation outcome showing the log-transformed value ofu y form e = and its Gaussian fit is shown in Fig. 3. Figure 4a shows how the mean value and standard deviation of Gaussian distributions fitted to ln(u y /m) are dependent on m e. While the experimental uncertainty has only a minor effect on the mean value of ln(u y /m), the standard deviation of the performance function is significantly influenced. An increase in the experimental uncertainty (increase of m e ) for the given total variability decreases the standard deviation of ln(u y /m). As explained earlier (recall Fig. 4), the experimental uncertainty affects the probability of unsatisfactory performance. Its value for natural variability and experimental uncertainty (denoted as p n ) for different values of m e is depicted in Fig. 4b. For the simulated problem, standard random field simulations ignoring experimental uncertainty (m e = ) yielded p n = 3.9% and simulations that considered experimental uncertainty with the current parameter set (m e = ) yielded p n =.5%. Thus, with increasing m e, the 2

23 Figure 2: Random fields of natural variability (m e = ) of the most influential parameters h s, n and β for one of the Monte-Carlo realisations. The bottom part of the mesh is not shown..2 Rel. frequency ln(u y / m) Figure 3: Histogram and log-normal fit of u y obtained in the Monte-Carlo simulation (m e =, 8 realisations). probability of unsatisfactory performance decreases to zero. The influence of sampling uncertainty on the probability of unsatisfactory performance To study the influence of sampling uncertainty the FOSM method was adopted, which was first evaluated with respect to full Monte-Carlo simulations. Figure 5 shows the results of a Monte- 22

24 .5 σ[ln(u y /m)] [-] µ[ln(u y /m)] [-] σ[ln(u y /m)] [-] µ[ln(u y /m)] [-] p n [%] m e [-] (a) m e [-] (b) Figure 4: (a) The dependency of the mean value and standard deviation of ln(u y /m) on m e ; (b) the dependency of the probability of unsatisfactory performance due to natural variability and experimental uncertainty p n on m e. Carlo simulation with 7 realisations for n t = 37, n e = 4 and m e =. The Monte-Carlo and FOSM simulation results practically coincide. Rel. frequency Monte-Carlo histogram Monte-Carlo Gaussian fit FOSM ln(u y /m) Figure 5: Comparison of Monte-Carlo and FOSM simulations revealing sampling uncertainty (n t = 37, n e = 4, m e = ). When σ s [Y] (standard deviation of the performance function due to sampling uncertainty) is calculated using the FOSM method, the total variance of Y can be calculated using Eq. (23), which can then be used in the evaluation of the probability of unsatisfactory performance due to natural variability and experimental and sampling uncertainties, denoted as p ns. In the present case of n t = 37, n e = 4 and m e =, p ns =.9%. When compared with p n =.5%, it can be seen that the sampling uncertainty increases the probability of unsatisfactory performance; for the current sample size the effect is, however, not substantial. The relatively small effect of sampling uncertainty on p ns is caused by the number of samples 23

25 n t = 37 and n e = 4, which is relatively high for standard practical projects. To understand the influence of n e and n t, two additional sets of FOSM simulations were run. In one set, n t was kept constant at a high value of n t = 8 and n e was varied. In the second set, n e = 8 and n t was varied. In the simulations, the values µ[x i ] and σ[ε n [X i ]] were kept constant. The results of these simulations are shown in Figs. 6a and b. n e has a small effect on p ns even at the relatively low value of n e = 5. In contrast, the effect of n t is more substantial, influencing simulation results significantly for n t < 4. This is because the tests on total variability are needed to specify both the mean value and variance of a parameter, wherease the tests on experimental uncertainty define variance only. It can be concluded that, in practical projects, an effort should be put into sampling for total variability, while the experimental uncertainty may be quantified using a relatively low number of samples. σ[ln(u y /m)] [-] n t variable, σ s [Y] n e variable, σ s [Y] n t variable, σ[y] n e variable, σ[y] p ns [%] n t variable n e variable n t or n e [-] (a) n t or n e [-] (b) Figure 6: (a) The dependency of the standard deviation of ln(u y /m) (sampling only σ s [Y] and total σ[y] ) on the number of samples; (b) the dependency of the probability of unsatisfactory performance p ns on the number of samples. Summary and conclusions An approach to quantify the effects of experimental and sampling uncertainties has been developed in this paper. The application of the proposed approach has been demonstrated by simulating the displacement of a foundation on a spatially variable sandy soil. Two extensive data sets were obtained by testing 37 samples revealing the effect of total variability and 4 samples revealing the effect of experimental uncertainty. It was demonstrated that the experimental uncertainty represents a substantial fraction of the measured total variability. Counter-intuitively, it has been shown that an increase in experimental uncertainty for the given total variability measured on spatially variable samples decreases the calculated probability of un- 24

26 satisfactory performance. It has been argued that this is possibly a reason for the relatively common overestimation of the probability of unsatisfactory performance in geotechnical engineering probabilistic simulations. By evaluating the sampling uncertainty it was found that while a relatively large number of samples (n t > 4) is needed to properly characterise the total variability, a much smaller number of samples (n e > 5) is sufficient for the experimental uncertainty quantification. Considering that no software adjustments are needed for the incorporation of experimental uncertainty into existing codes based on random field methods, and that relatively little additional experimental effort is needed for experimental uncertainty quantification, it is recommended that a consideration of experimental uncertainty should become more common in geotechnical engineering probabilistic analyses. Acknowledgment Financial support by the research grant GACR P5/2/75 is greatly appreciated. The tests quantifying the experimental uncertainty were performed by P. Mayer within his MSc project. The author is grateful to doc. J. Ježek, doc. Z. Hlávka and to J. Ondřej for valuable discussions on the manuscript. References Akbas, S. O. & Kulhawy, F. H. (2). Characterisation and estimation of geotechnical variability in Ankara clay: A case history. Geotech Geol Eng 28, Al-Naqshabandy, M. S. & Larsson, S. (23). Effect of uncertainties of improved soil shear strength on the reliability of embankments. Journal of Geotechnical and Geoenvironmental Engineering ASCE 39, No. 4, Andrade, J. E., Baker, J. W. & Ellison, K. C. (28). Random porosity fields and their influence on the stability of granular media. International Journal for Numerical and Analytical Methods in Geomechanics 32, Bauer, E. (996). Calibration of a comprehensive constitutive equation for granular materials. Soils and Foundations 36, No., Booy, C. & Lye, L. M. (989). A new look at flood risk determination. Water Resources Bulletin 25, No. 5, Bourdeau, P. L. & Amundaray, J. I. (25). Non-parametric simulation of geotechnical variability. Géotechnique 55, No. 2,